Cordoba, who haven’t won since January, held out for 42 minutes, but a four-goal blitz in 11 minutes either side of half-time ended their return to the top flight after just one season.
Ivan Rakitic smashed home the opener in style before Suarez added the second with the final touch of the first-half.
Lionel Messi, Suarez and Gerard Pique all then headed home before Messi had the simplest of finishes to roll in his 40th league goal of the season to move one ahead of Cristiano Ronaldo as the league’s top goalscorer.
Messi then allowed Neymar to convert from the penalty spot before Suarez completed his hat-trick two minutes from time.
Victory takes Barca five points clear of Real Madrid, who face a tough trip to Sevilla later on Saturday.
Cordoba could yet be handed a reprieve from relegation should Almeria lose their appeal against the deduction of three points for unpaid transfer fees, or Elche face sanctions for overdue debts to the tax authorities.
“Until the first goal we had created a lot of chances and were fearing it wouldn’t be our day,” said Barca boss Luis Enrique.
“The goals from Ivan and Luis helped us and in the second-half we were very effective in front of goal. I am happy because they are three very important points.”
Despite Wednesday’s looming meeting with former Barca boss Pep Guardiola and German champions Bayern Munich in their Champions League semi-final, first-leg, Enrique named his strongest available side.
Neymar wasted a great chance to get the visitors off to a flying start when he volleyed over from Alves’s cross.
Cordoba missed a similar opening moments later as Florin Andone headed over Edimar’s dangerous ball in from the left.
Messi also uncharacteristically passed up a great opportunity to open the scoring as he shot straight at Juan Carlos after brilliantly dribbling past three defenders, whilst another Neymar mishit shot came back off the post.
However, it was only a matter of time before the league leaders made the breakthrough and they did so in style when Messi’s lovely cushioned pass picked out Rakitic and he crashed a shot into the roof of the net.
Suarez made it 2-0 just before the break when he latched onto a similar pass from Andres Iniesta to sneak the ball between Carlos’s legs.
The one way traffic continued after the break as Messi expertly nodded home Alves’s inch-perfect cross just a minute into the second-half.
Alves’s brilliant form in recent weeks has been unimpeded by ongoing speculation over his future as he once more bombed to the right to cross for Suarez to head in his 23rd goal of the season.
Neymar’s off-day in front of goal continued when he sliced wide with just the keeper to beat from Suarez’s inviting pass.
However, Barca didn’t have to wait long for a fifth as Pique rose unchallenged to head in from Xavi’s corner 25 minutes from time.
Messi rolled home the sixth into an empty net thanks to unselfish play by Neymar and the four-time World Player of the Year returned the favour by allowing the Brazilian to slot home from the penalty spot after he had been felled by Adrian Gunino.
And Suarez rounded off the scoring by completing his first Barca hat-trick with a simple finish after Xavi’s blocked effort had left the helpless Carlos on the floor.